It's tough to find a small rifle these days. In the past the CVA Squirrel Rifle and T/Cs Seneca and Cherokee were available for kids and small adults.
My wife is five foot and weighs about 100 pounds. When we were starting out about the smallest available was a CVA Kentucky rifle. I got a kit for her and took six inches off the muzzle and reduced the pull by one and half inches. She shot that for about five years before a fire took it.
Whatever you decide to do, build one from scratch or reduce the size of an existing kit, be aware that kids grow faster than you think. Make it a little on the big size to fit him now. He will be using it from the bench or off cross-sticks for a while and will grow into it. After he hits his teen years, you will probably have to restock it anyway or sell it to someone who needs it for his kid.